Samson Abramsky
Temperley-Lieb Algebra:
From Knot Theory to
Logic and Computation
via Quantum Mechanics
1
Introduction
Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of
apparently disparate topics.
1.1
Knot Theory
Vaughan Jones’ discovery of his new polynomial invariant of knots in
1984 [26] triggered a spate of mathematical developments relating knot
theory, topological quantum field theory, and statistical physics inter
alia [44, 30]. A central rôle, both in the initial work by Jones and in the
subsequent developments, was played by what has come to be known as
the Temperley-Lieb algebra.1
1.2
Categorical Quantum Mechanics
Recently, motivated by the needs of Quantum Information and Computation, Abramsky and Coecke have recast the foundations of Quantum
Mechanics itself, in the more abstract language of category theory. The
1 The
original work of Temperley and Lieb [43] was in discrete lattice models of
statistical physics. In finding exact solutions for a certain class of systems, they had
identified the same relations which Jones, quite independently, found later in his
work.
key contribution is the paper [4], which develops an axiomatic presentation of quantum mechanics in the general setting of strongly compact
closed categories, which is adequate for the needs of Quantum Information and Computation. Moreover, this categorical axiomatics can be
presented in terms of a diagrammatic calculus which is both intuitive
and effective, and can replace low-level computation with matrices by
much more conceptual reasoning. This diagrammatic calculus can be
seen as a proof system for a logic [6], leading to a radically new perspective on what the right logical formulation for Quantum Mechanics
should be.
This line of work has a direct connection to the Temperley-Lieb algebra, which can be put in a categorical framework, in which it can be
described essentially as the free pivotal dagger category on one self-dual
generator [21].2 Here pivotal dagger category is a non-symmetric (“planar”) version of (strongly or dagger) compact closed category — the key
notion in the Abramsky-Coecke axiomatics.
1.3
Logic and Computation
The Temperley-Lieb algebra itself has some direct and striking connections to basic ideas in Logic and Computation, which offer an intriguing
and promising bridge between these prima facie very different areas. We
shall focus in particular on the following two topics:
• The Temperley-Lieb algebra has always hitherto been presented
as a quotient of some sort: either algebraically by generators and
relations as in Jones’ original presentation [26], or as a diagram
algebra modulo planar isotopy as in Kauffman’s presentation [29].
We shall use tools from Geometry of Interaction [23], a dynamical interpretation of proofs under Cut Elimination developed as
an off-shoot of Linear Logic [22], to give a direct description of
the Temperley-Lieb category — a fully abstract presentation, in
Computer Science terminology [37]. This also brings something
new to the Geometry of Interaction, since we are led to develop a
planar version of it, and to verify that the interpretation of CutElimination (the “Execution Formula” [23], or “composition by
feedback” [8, 1]) preserves planarity.
2 Strictly
speaking, the full Temperley-Lieb category over a ring R is the free R-linear
enrichment of this free pivotal dagger category.
• We shall also show how the Temperley-Lieb algebra provides a
natural setting in which computation can be performed diagrammatically as geometric simplification — “yanking lines straight”.
We shall introduce a “planar λ-calculus” for this purpose, and
show how it can be interpreted in the Temperley-Lieb category.
1.4
Outline of the Paper
We briefly summarize the further contents of this paper. In Section 2
we introduce the Temperley-Lieb algebras, emphasizing Kauffman’s diagrammatic formulation. We also briefly outline how the Temperley-Lieb
algebra figures in the construction of the Jones polynomial. In Section 3 we describe the Temperley-Lieb category, which provides a more
structured perspective on the Temperley-Lieb algebras. In Section 4, we
discuss some features of this category, which have apparently not been
considered previously, namely a characterization of monics and epics,
leading to results on image factorization and splitting of idempotents. In
Section 5, we briefly discuss the connections with the Abramsky-Coecke
categorical formulation of Quantum Mechanics, and raise some issues
and questions about the possible relationship betwen planar, braided
and symmetric settings for Quantum Information and Computation. In
Section 6 we develop a planar version of Geometry of Interaction, and
the direct “fully abstract” presentation of the Temperley-Lieb category.
In Section 7 we discuss the planar λ-calculus and its interpretation in the
Temperley-Lieb category. We conclude in Section 8 with some further
directions.
Note to the Reader Since this paper aims at indicating crosscurrents between several fields, it has been written in a somewhat expansive style, and an attempt has been made to explain the context of
the various ideas we will discuss. We hope it will be accessible to readers
with a variety of backgrounds.
2
The Temperley-Lieb Algebra
Our starting point is the Temperley-Lieb algebra, which has played a
central role in the discovery by Vaughan Jones of his new polynomial
invariant of knots and links [26], and in the subsequent developments
over the past two decades relating knot theory, topological quantum
field theory, and statistical physics [30].
Jones’ approach was algebraic: in his work, the Temperley-Lieb algebra was originally presented, rather forbiddingly, in terms of abstract
generators and relations. It was recast in beautifully elementary and
conceptual terms by Louis Kauffman as a planar diagram algebra [29].
We begin with the algebraic presentation.
2.1
Temperley-Lieb algebra: generators and relations
We fix a ring R; in applications to knot polynomials, this is taken to be
a ring of Laurent polynomials C[X, X −1 ]. Given a choice of parameter
τ ∈ R and a dimension n ∈ N, we define the Temperley-Lieb algebra
An (τ ) to be the unital, associative R-linear algebra with generators
U1 , . . . , Un−1
and relations
Ui Uj Ui = Ui
Ui2
|i − j| = 1
= τ · Ui
Ui Uj = Uj Ui
|i − j| > 1
Note that the only relations used in defining the algebra are multiplicative ones. This suggests that we can obtain the algebra An (τ ) by
presenting the multiplicative monoid Mn , and then P
obtaining An (τ ) as
the monoid algebra of formal R-linear combinations i ri · ai over Mn ,
with the multiplication in An (τ ) defined as the bilinear extension of the
monoid multiplication in Mn :
X
X
X
(ri sj ) · (ai bj ).
sj · b j ) =
ri · ai )(
(
i
i,j
j
We define Mn as the monoid with generators
δ, U1 , . . . , Un−1
and relations
Ui Uj Ui = Ui
Ui2
|i − j| = 1
= δUi
Ui Uj = Uj Ui
δUi = Ui δ
|i − j| > 1
We can then obtain An (τ ) as the monoid algebra over Mn , subject to
the identification
δ = τ · 1.
2.2
Diagram Monoids
These formal algebraic ideas are brought to vivid geometric life by
Kauffman’s interpretation of the monoids Mn as diagram monoids.
We start with two parallel rows of n dots (geometrically, the dots are
points in the plane). The general form of an element of the monoid is
obtained by “joining up the dots” pairwise in a smooth, planar fashion,
where the arc connecting each pair of dots must lie within the rectangle
framing the two parallel rows of dots. Such diagrams are identified up
to planar isotopy, i.e. continuous deformation within the portion of the
plane bounded by the framing rectangle..
Thus the generators U1 , . . . , Un−1 can be drawn as follows:
1 2 3
···
n
1
···
n
···
1 2 3
···
U1
n
1
···
n
Un−1
The generator δ corresponds to a loop — all such loops are identified
up to isotopy.
We refer to arcs connecting dots in the top row as cups, those connecting dots in the bottom row as caps, and those connecting a dot in
the top row to a dot in the bottom row as through lines.
Multiplication xy is defined by identifying the bottom row of x with
the top row of y, and composing paths. In general loops may be formed
— these are “scalars”, which can float freely across these figures. The
relations can be illustrated as follows:
=
U1 U2 U1 = U1
=
U12 = δU1
=
U1 U3 = U3 U1
2.3
Expressiveness of the Generators
The fact that all planar diagrams can be expressed as products of
generators is not entirely obvious. For proofs, see [29, 20]. As an illustrative example, consider the planar diagrams in M3 . Apart from the
generators U1 , U2 , and ignoring loops, there are three:
The first is the identity for the monoid; we refer to the other two as the
left wave and right wave respectively. The left wave can be expressed as
the product U2 U1 :
=
The right wave has a similar expression.
Once we are in dimension four or higher, we can have nested cups and
caps. These can be built using waves, as illustrated by the following:
=
2.4
The Trace
There is a natural trace function on the Temperley-Lieb algebra, which
can be defined diagrammatically on Mn by connecting each dot in the
top row to the corresponding dot in the bottom row, using auxiliary
cups and cups. This always yields a diagram isotopic to a number of
loops — hence to a scalar, as expected. This trace can then be extended
linearly to An (τ ).
We illustrate this firstly by taking the trace of a wave—which is equal
to a single loop:
=
The Ear is a Circle
Our second example illustrates the important general point that the
trace of the identity in Mn is δ n :
=
2.5
The Connection to Knots
How does this connect to knots? Again, a key conceptual insight is due
to Kauffman, who saw how to recast the Jones polynomial in elementary
combinatorial form in terms of his bracket polynomial. The basic idea of
the bracket polynomial is expressed by the following equation:
=
A
+
B
Each over-crossing in a knot or link is evaluated to a weighted sum
of the two possible planar smoothings. With suitable choices for the
coefficients A and B (as Laurent polynomials), this is invariant under
the second and third Reidemeister moves. With an ingenious choice
of normalizing factor, it becomes invariant under the first Reidemeister
move — and yields the Jones polynomial! What this means algebraically
is that the braid group Bn has a representation in the Temperley-Lieb
algebra An (τ ) — the above bracket evaluation shows how the generators
βi of the braid group are mapped into the Temperley-Lieb algebra:
βi 7→ A · Ui + B · 1.
Every knot arises as the closure (i.e. the diagrammatic trace) of a braid;
the invariant arises by mapping the open braid into the Temperley-Lieb
algebra, and taking the trace there.
This is just the beginning of a huge swathe of further developments,
including: Topological Quantum Field Theories [44], Quantum Groups
[28], Quantum Statistical mechanics [30], Diagram Algebras and Representation Theory [25], and more.
3
The Temperley-Lieb Category
We can expose more structure by gathering all the Temperley-Lieb
algebras into a single category. We begin with the category D which
plays a similar role with respect to the diagram monoids Mn .
The objects of D are the natural numbers. An arrow n → m is given
by
• a number k ∈ N of loops
• a diagram which joins the top row of n dots and the bottom row of
m dots up pairwise, in the same smooth planar fashion as we have
already specified for the diagram monoids. As before, diagrams
are identified up to planar isotopy.
Composition of arrows f : n → m and g : m → p is defined by identifying the bottom row of m dots for f with the top row of m dots for g,
and composing paths. The loops in the resulting arrow are those of f
and of g, together with any formed by the process of composing paths.
Clearly we recover each Mn as the endomorphism monoid D(n, n).
Moreover, we can define the Temperley-Lieb category T over a ring R
as the free R-linear category generated by D, with a construction which
generalizes that of the monoid algebra: the objects of T are the same as
those of D, and arrows are R-linear combinations of arrows of D, with
composition defined by bilinear extension from that in D:
X
X
X
(ri sj ) · (gi ◦ fj ).
sj · f j ) =
ri · gi ) ◦ (
(
i
j
i,j
If we fix a parameter τ ∈ R, then we obtain the category Tτ by the
identification of the loop in D with the scalar τ in T .3 We then
recover the Temperley-Lieb algebras as
An (τ ) = Tτ (n, n).
New possibilities also arise in D. In particular, we get the pure cap
as (the unique) arrow 0 → 2, and similarly the pure cup as the unique
arrow 2 → 0. More generally, for each n we have arrows ηn : 0 → n + n,
and ǫn : n + n → 0:
1 ...
1
...
...
. . . 2n
2n
We refer to the arrows ηn as units, and the arrows ǫn as counits.
The category D has a natural strict monoidal structure. On objects,
we define n ⊗ m = n + m, with unit given by I = 0. The tensor product
of morphisms
f :n→m g:p→q
f ⊗g :n+p→p+q
is given by juxtaposition of diagrams in the evident fashion, with (multiset) union of loops. Thus we can write the units and counits as arrows
ηn : I → n ⊗ n,
ǫn : n ⊗ n → I.
These units and counits satisfy important identities, which we illustrate
diagrammatically
=
=
and write algebraically as
(ǫn ⊗ 1n ) ◦ (1n ⊗ ηn ) = 1n = (1n ⊗ ǫn ) ◦ (ηn ⊗ 1n ).
3 The
(1)
full justification of this step requires the identification of D as a free pivotal
category, as discussed below.
3.1
Pivotal Categories
From these observations, we see that D is a strict pivotal category
[21], in which the duality on objects is trivial: A = A∗ . We recall that
a strict pivotal category is a strict monoidal category (C, ⊗, I) with an
assignment A 7→ A∗ on objects satisfying
A∗∗ = A,
(A ⊗ B)∗ = B ∗ ⊗ A∗ ,
I ∗ = I,
and for each object A, arrows
ηA : I → A∗ ⊗ A,
ǫA : A ⊗ A∗ → I
satisfying the triangular identities:
(ǫA ⊗ 1A ) ◦ (1A ⊗ ηA ) = 1A ,
(1A∗ ⊗ ǫA ) ◦ (ηA ⊗ 1A∗ ) = 1A∗ .
In addition, the following coherence equations are required to hold:
ηA⊗B = (1B ∗ ⊗ ηA ⊗ 1B ) ◦ ηB ,
ηI = 1I ,
and, for f : A → B:
1 ⊗ f ⊗1 ∗
A ⊗ B ⊗ B∗
⊗
⊗
1
1
-
A∗ ⊗ A ⊗ B ∗
ηA
ǫB
-
B∗
-
A∗
1
1
⊗
⊗
-
∗
ǫB
∗
ηA
B ∗ ⊗ A ⊗ A∗
- B ∗ ⊗ B ⊗ A∗
1⊗f ⊗1
This last equation is illustrated diagrammatically by
f
=
f
(2)
We extend ()∗ to a contravariant involutive functor:
f :A→B
f ∗ : B ∗ → A∗
f ∗ = (1 ⊗ ǫA ) ◦ (1 ⊗ f ⊗ 1) ◦ (ηA ⊗ 1)
which indeed satisfies
1∗ = 1,
(g ◦ f )∗ = f ∗ ◦ g ∗ ,
f ∗∗ = f,
the last equation being illustrated diagrammatically by
A
A∗
A
A
=
f
B
B∗
f
B
B
These axioms have powerful consequences. In particular, C is monoidal
closed, with internal hom given by A∗ ⊗ B, and the adjunction:
C(A ⊗ B, C) ≃ C(B, A∗ ⊗ C) :: f 7→ (1 ⊗ f ) ◦ (ηA ⊗ 1).
This means that a restricted form of λ-calculus can interpreted in such
categories — a point we shall return to in Section 7.
A trace function can be defined in pivotal categories, which takes an
endmorphism f : A → A to a scalar in C(I, I):
Tr(f ) = ǫA ◦ (f ⊗ 1) ◦ ηA∗ .
It satisfies:
Tr(g ◦ f ) = Tr(f ◦ g).
In D, this definition yields exactly the diagrammatic trace we discussed
previously.
We have the following important characterization of the diagrammatic
category D:
PROPOSITION 1.1
D is the free pivotal category over one self-dual generator; that is, freely
generated over the one-object one-arrow category, with object A say,
subject to the equation A = A∗ .
This was mentioned (although not proved) in [21]; see also [20]. The
methods in [3] can be adapted to prove this result, using the ideas we
shall develop in Section 6.
The idea of “identifying the loop with the scalar τ ” in passing from
D to the full Temperley-Lieb category Tτ can be made precise using the
construction given in [3] of gluing a specified ring R of scalars onto a
free compact closed category, along a given map from the loops in the
generating category to R. In this case, there is a single loop in the
generating category, and we send it to τ .
3.2
Pivotal Dagger Categories
We now mention a strengthening of the axioms for pivotal categories,
corresponding to the notion of strongly compact closed or dagger compact
closed category which has proved to be important in the categorical approach to Quantum Mechanics [4, 5]. Again we give the strict version for
simplicity. We assume that the strict monoidal category (C, ⊗, I) comes
equipped with an identity-on-objects, contravariant involutive functor
†
†
()† such that ǫA = ηA
abstracts from the adjoint
∗ . The idea is that f
of a linear map, and allows the extra structure arising from the use of
complex Hilbert spaces in Quantum Mechanics to be expressed in the
abstract setting.
Note that there is a clear diagrammatic distinction between the dual
f ∗ and the adjoint f † . The dual corresponds to 180◦ rotation in the
plane:
A1
A
··· n
∗
Bm
B∗
··· 1
f∗
f
B1
···
Bm
A∗n
while the adjoint is reflection in the x-axis:
··· ∗
A1
A1
A
··· n
B1
f†
f
B1
B
··· n
···
Bm
A1
···
An
For example in D, if we consider the left and right wave morphisms L
and R:
then we have
L∗ = L,
L† = R,
R∗ = R,
R† = L.
Using the adjoint, we can define a covariant functor
f :A→B
f∗ : A∗ → B ∗
f 7→ f ∗† .
We have
(f ∗ )∗ = f † = (f∗ )∗ .
In terms of complex matrices, f ∗ is transpose, while f∗ is complex conjugation. Diagrammatically, f∗ is “reflection in the y-axis”.
f
f∗
f†
f∗
We have the following refinement of Proposition 1.1, by similar methods to those used for free strongly compact closed categories in [3].
PROPOSITION 1.2
D is the free pivotal dagger category over one self-dual generator.
4
Factorization and Idempotents
We now consider some structural properties of the category D which
we have not found elsewhere in the literature.4
We begin with a pleasingly simple diagrammatic characterization of
monics and epics in D.
PROPOSITION 1.3
An arrow in D is monic iff it has no cups; it is epic iff it has no caps.
Proof
Suppose that f : n → m has no cups. Thus all dots in n are
connected by through lines to dots in m. Now consider a composition
f ◦ g. No loops can be formed by this composition; hence we can recover
g from f ◦ g by erasing the caps of f . Moreover, the number of loops in
f ◦ g will simply be the sum of the loops in f and g, so we can recover
the loops of g by subtracting the loops of f from the composition. It
follows that
f ◦ g = f ◦ h =⇒ g = h,
i.e. that f is monic, as required.
For the converse, suppose that f has a cup, which we can assume
to be connecting dots i and i + 1 in the top row. (Note that if i < j
are connected by a cup, then by planarity, every k with i < k < j
must also be connected in a cup to some l with i < l < j.) Then
f ◦ δ · 1 = f ◦ (1 ⊗ Ui ⊗ 1), so f is not monic. Diagrammatically, this
says that we can either form a loop using the cup of f , or simply add a
loop which is attached to an identity morphism.
The characterization of epics is entirely similar.
This immediately yields an “image factorization” structure for D.
PROPOSITION 1.4
Every arrow in D has an epi-mono factorization.
Proof Given an arrow f : n → m, suppose it has p cups and q caps.
Then we obtain arrows e : n → (m − 2q) by erasing the caps, and
m : (n − 2p) → m by erasing the cups. By Proposition 1.3, e is epic and
4 The
idea of considering these properties arose from a discussion with Louis Kauffman, who showed the author a direct diagrammatic characterization of idempotents
in D, which has subsequently appeared in [32].
m monic. Moreover, the number of dots in the top and bottom rows
connected by through lines must be the same. Hence
(m − 2q) = k = (n − 2p),
and we can compose e and m to recover f . Note that by planarity, once
we have assigned cups and caps, there is no choice about the correspondence between top and bottom row dots by through lines.
This factorization is “essentially” unique. However, we are free to
split the l loops of f between e and m in any way we wish, so there is a
distinct factorization δ a · m ◦ δ b · e for all a, b ∈ N with a + b = l.
We illustrate the epi-mono factorization for the left wave:
=
We recall that an idempotent in a category is an arrow i : A → A
such that i2 = i. We say that an idempotent i splits if there are arrows
r : A → B and s : B → A such that
i = s ◦ r,
r ◦ s = 1B .
PROPOSITION 1.5
All idempotents split in D.
Proof
Let i : n → n be an idempotent in D. By Proposition 1.4,
i = m ◦ e, where e : n → k is epic and m : k → n is monic. Now
m ◦ e ◦ m ◦ e = m ◦ e.
Since m is monic, this implies that e ◦ m ◦ e = e = 1 ◦ e. Since e is epic,
this implies that e ◦ m = 1.
5
Categorical Quantum Mechanics
We now relate our discussion to the Abramsky-Coecke programme of
Categorical Quantum Mechanics.
This approach is very different to previous work on the Computer Science side of this interdisciplinary area, which has focussed on quantum
algorithms and complexity. The focus has rather been on developing
high-level methods for Quantum Information and Computation (QIC)—
languages, logics, calculi, type systems etc.—analogous to those which
have proved so essential in classical computing [2]. This has led to nothing less than a recasting of the foundations of Quantum Mechanics itself,
in the more abstract language of category theory. The key contribution
is the paper with Coecke [4], in which we develop an axiomatic presentation of quantum mechanics in the general setting of strongly compact
closed categories, which is adequate for all the needs of QIC.
Specifically, we show that we can recover the key quantum mechanical notions of inner-product, unitarity, full and partial trace, HilbertSchmidt inner-product and map-state duality, projection, positivity, measurement, and Born rule (which provides the quantum probabilities), axiomatically at this high level of abstraction and generality. Moreover,
we can derive the correctness of protocols such as quantum teleportation, entanglement swapping and logic-gate teleportation [10, 24, 45] in
a transparent and very conceptual fashion. Also, while at this level of
abstraction there is no underlying field of complex numbers, there is still
an intrinsic notion of ‘scalar’, and we can still make sense of dual vs. adjoint [4, 5], and global phase and elimination thereof [15]. Peter Selinger
recovered mixed state, complete positivity and Jamiolkowski map-state
duality [42]. Recently, in collaboration with Dusko Pavlovic and Eric Paquette, decoherence, generalized measurements and Naimark’s theorem
have been recovered [17, 16].
Moreover, this formalism has two important additional features. Firstly,
it goes beyond the standard Hilbert-space formalism, in that it is able to
capture classical as well as quantum information flows, and the interaction between them, within the formalism. For example, we can capture
the idea that the result of a measurement is used to determine a further
stage of quantum evolution, as e.g. in the teleportation protocol [10],
where a unitary correction must be performed after a measurement; or
also in measurement-based quantum computation [39, 40]. Secondly,
this categorical axiomatics can be presented in terms of a diagrammatic
calculus which is extremely intuitive, and potentially can replace lowlevel computation with matrices by much more conceptual — and automatable — reasoning. Moreover, this diagrammatic calculus can be
seen as a proof system for a logic, leading to a radically new perspective
on what the right logical formulation for Quantum Mechanics should
be. This latter topic is initiated in [6], and developed further in the
forthcoming thesis of Ross Duncan.
5.1
Outline of the approach
We now give some further details of the approach. The general setting
is that of strongly (or dagger) compact closed categories, which are the
symmetric version of the pivotal dagger categories we encountered in
Section 3. Thus, in addition to the structure mentioned there, we have
a symmetry natural isomorphism
σA,B : A ⊗ B ≃ B ⊗ A.
See [5] for an extended discussion. An important feature of the AbramskyCoecke approach is the use of an intuitive graphical calculus, which is
essentially the diagrammatic formalism we have seen in the TemperleyLieb setting, extended with more general basic types and arrows. The
key point is that this formalism admits a very direct physical interpretation in Quantum Mechanics.
In the graphical calculus we depict physical processes by boxes, and we
label the inputs and outputs of these boxes by types which indicate the
kind of system on which these boxes act, e.g. one qubit, several qubits,
classical data, etc. Sequential composition (in time) is depicted by connecting matching outputs and inputs by wires, and parallel composition
(tensor) by locating entities side by side e.g.
1A : A → A
f :A→B
g ◦f
1A ⊗1B
f ⊗1C
f ⊗g
(f ⊗g)◦h
for g : B → C and h : E → A ⊗ B are respectively depicted as:
C
B
A
g
f
B
A
f
A
C
B
B
A
B
f
A
B
C
f
A
C
g
B
g
f
B
A
h
E
(The convention in these diagrams is that the ‘upward’ vertical direction
represents progress of time.) A special role is played by boxes with either
no input or no output, called states and costates respectively (cf. Dirac’s
kets and bras [19]) which we depict by triangles. Finally, we also need to
consider diamonds which arise by post-composing a state with a matching costate (cf. inner-product or Dirac’s bra-ket):
π oψ
A
π
=
π
A
ψ
A
ψ
that is, algebraically,
ψ:I→A
π:A→I
π◦ψ :I→I
where I is the tensor unit : A ⊗ I ≃ A ≃ I ⊗ A. Extra structure is
represented by (i) assigning a direction to the wires, where reversal of
this direction is denoted by A 7→ A∗ , (ii) allowing reversal of boxes
(cf. the adjoint for vector spaces), and, (iii) assuming that for each type
A there exists a special bipartite Bell-state and its adjoint Bell-costate:
A
B
A
A*
f†
f
A*
A
A*
B
A
A
that is, algebraically,
A
A∗
f† : B → A
f :A→B
†
ηA
: A∗ ⊗A → I.
ηA : I → A∗ ⊗A
Hence, bras and kets are adjoint and the inner product has the form
(−)† ◦ (−) on states. Essentially the sole axiom we impose is:
A
A*
A
=
A
that is, algebraically,
†
(ηA
∗ ⊗ 1A ) ◦ (1A ⊗ ηA ) = 1A .
If we extend the graphical notation of Bell-(co)states to:
A*
A
A*
A
we obtain a clear graphical interpretation for the axiom:
=
(1)
which now tells us that we are allowed to yank the black line straight :
=
This equation and its diagrammatic counterpart should of course be
compared to equation (2), and equation (1) and its accompanying diagram, in Section 3— they are one and the same, subject to minor
differences in diagrammatic conventions.
This intuitive graphical calculus is an important benefit of the categorical axiomatics. Other advantages can be found in [4, 2].
5.2
Quantum non-logic vs. quantum hyper-logic
The term quantum logic is usually understood in connection with the
1936 Birkhoff-von Neumann proposal [11, 41] to consider the (closed)
linear subspaces of a Hilbert space ordered by inclusion as the formal expression of the logical distinction between quantum and classical physics.
While in classical logic we have deduction, the linear subspaces of a
Hilbert space form a non-distributive lattice and hence there is no obvious notion of implication or deduction. Quantum logic was therefore
always seen as logically very weak, or even as a non-logic. In addition, it has never given a satisfactory account of compound systems and
entanglement.
On the other hand, compact closed logic in a sense goes beyond ordinary logic in the principles it admits. Indeed, while in ordinary categorical logic “logical deduction” implies that morphisms internalize as
elements (which above we referred to above as states) i.e.
B
f
- C
≃
←→
I
⌈f ⌉
- B ⇒C
(where I is the tensor unit), in compact closed logic they internalize both
as states and as costates, i.e.
A ⊗ B∗
⌊f ⌋
- I
≃
←→
A
f
- B
≃
←→
I
⌈f ⌉
- A∗ ⊗ B
where we introduce the following notation:
pf q = (1A∗ ⊗ f ) ◦ ηA : I → A∗ ⊗ B
xf y = ǫB ◦ (f ⊗ 1B ∗ ) : A ⊗ B ∗ → I.
It is exactly this dual internalization which allows the straightening axiom in picture (1) to be expressed. In the graphical calculus this is
witnessed by the fact that we can define both a state and a costate
f
=:
=:
f
f
f
(2)
for each operation f . Physically, costates form the (destructive parts of)
projectors, i.e. branches of projective measurements.
5.2.1
Compositionality.
The semantics is obviously compositional, both with respect to sequential composition of operations and parallel composition of types
and operations, allowing the description of systems to be built up from
smaller components. But we also have something more specific in mind:
a form of compositionality with direct applications to the analysis of
compound entangled systems. Since we have:
g
g
=
=
g
f
=
f
g
f
f
we obtain:
g
=
f
g
f
(3)
i.e. composition of operations can be internalized in the behavior of
entangled states and costates. Note in particular the interesting phenomenon of “apparant reversal of the causal order” which is the source
of many quite mystical interpretations of quantum teleportation in terms
of “traveling backward in time” — cf. [35]. Indeed, while on the left,
physically, we first prepare the state labeled g and then apply the costate
labeled f , the global effect is as if we first applied f itself first, and only
then g.
5.2.2
Derivation of quantum teleportation.
This is the most basic application of compositionality in action. Immediately from picture (1) we can read the quantum mechanical potential
for teleportation:
ψ
=
=
ψ
ψ
Alice
Bob
Alice
Alice
Bob
Bob
This is not quite the whole story, because of the non-deterministic nature
of measurements. But it suffices to introduce a unitary correction. Using
picture (3) the full description of teleportation becomes:
=
fi
-1
fi
fi
=
-1
fi
where the classical communication is now implicit in the fact that the
index i is both present in the costate (= measurement-branch) and the
correction, and hence needs to be sent from Alice to Bob.
The classical communication can be made explicit as a fully fledged
part of the formalism, using additional types: biproducts in [4], and
“classical objects” in [17]. This allows entire protocols, including the
interplay between quantum and classical information which is often their
most subtle ingredient, to be captured and reasoned about rigorously in
a single formal framework.
5.3
Remarks
We close this Section with some remarks. We have seen that the categorical and diagrammatic setting for Quantum Mechanics developed by
Abramsky and Coecke is strikingly close to that in which the TemperleyLieb category lives. The main difference is the free recourse to symmetry
allowed in the Abramsky-Coecke setting (and in the main intended models for that setting, namely finite-dimensional Hilbert spaces with linear
or completely positive maps). However, it is interesting to note that
in the various protocols and constructions in Quantum Information and
Computation which have been modelled in that setting to date [4], the
symmetry has not played an essential role. The example of teleportation
given above serves as an example.
This raises some natural questions:
How much of QM/QIC lives in the plane?
More precisely:
• Which protocols make essential use of symmetry?
• How much computational or information-processing power does
the non-symmetric calculus have?
• Does braiding have some computational significance? (Remembering that between pivotal and symmetric we have braided strongly
compact closed categories) [21].
6
Planar Geometry of Interaction and the TemperleyLieb Algebra
We now address the issue of giving what, so far as I know, is the first
direct —or “fully abstract”—description of the Temperley-Lieb category.
Since the category T is directly and simply described as the free R-linear
category generated by D, we focus on the direct description of D.
Previous descriptions:
• Algebraic, by generators and relations - whether “locally”, of the
Temperley-Lieb algebras An (τ ), as in Jones’ presentation, or “globally”, by a description of D as the free pivotal category, as in
Proposition 1.1.
• Kauffman’s topological description: diagrams “up to planar isotopy”.
In fact, it is well known (see e.g. [33]) that the diagrams are completely
characterized by how the dots are joined up — i.e. by discrete relations
on finite sets. This leaves us with the problem of how to capture
1. Planarity
2. The multiplication of diagrams — i.e. composition in D
purely in terms of the data given by these relations.
The answers to these questions exhibit the connections that exist between the Temperley-Lieb category and what is commonly known as
the “Geometry of Interaction”. This is a dynamical/geometrical interpretation of proofs and Cut Elimination initiated by Girard [23] as an
off-shoot of Linear Logic [22]. The general setting for these notions is
now known to be that of traced monoidal and compact closed categories
— in particular, in the free construction of compact closed categories
over traced monoidal categories [1, 7]. In fact, this general construction
was first clearly described in [27], where one of the leading motivations
was the knot-theoretic context.
Our results in this Section establish a two-way connection. In one
direction, we shall use ideas from Geometry of Interaction to answer
Question 2 above: that is, to define path composition (including the formation of loops) purely in terms of the discrete relations tabulating how
the dots are joined up. In the other direction, our answer to Question 1
will allow us to consider a natural planar variant of the Geometry of
Interaction.
6.1
6.1.1
Some preliminary notions
Partial Orders
We use the notation P = (|P |, ≤P ) for partial orders. Thus |P | is
the underlying set, and ≤P is the order relation (reflexive, transitive
and antisymmetric) on this set. An order relation is linear if for all
x, y ∈ |P |, x ≤P y or y ≤P x.
Given a natural number n, we define [n] := {1 < · · · < n}, the linear
order of length n. We define several constructions on partial orders.
Given partial orders P , Q, we define:
• The disjoint sum P ⊕ Q, where |P ⊕ Q| = |P | + |Q|, the disjoint
union of |P | and |Q|, and
x ≤P ⊕Q y ⇐⇒ (x ≤P y) ∨ (x ≤Q y).
• The concatenation P Q, where |P Q| = |P | + |Q|, with the
following order:
x ≤P Q y ⇐⇒ (x ≤P y) ∨ (x ≤Q y) ∨ (x ∈ P ∧ y ∈ Q).
• P op = (|P |, ≥P ).
Given elements x, y of a partial order P , we define:
x↑y
⇔ (x ≤P y) ∨ (y ≤P x)
x#y ⇔
6.1.2
¬(x ↑ y).
Relations
A relation on a set X is a subset of the cartesian product: R ⊆ X ×X.
Since relations are sets, they are closed under unions and intersections.
We shall also use the following operations of relation algebra:
Identity relation:
Relation composition:
1X := {(x, x) | x ∈ X}
R; S := {(x, z) | ∃y. (x, y) ∈ R ∧ (y, z) ∈ S}
Rc := {(y, x) | (x, y) ∈ R}
S
Transitive closure:
R+ := k≥1 Rk
S
Reflexive transitive closure: R∗ := k≥0 Rk
Relational converse:
Here Rk is defined inductively: R0 := 1X , R1 := R, Rk+1 := R; Rk . A
relation R is single-valued or a partial function if Rc ; R ⊆ 1X . It is total
if R; Rc ⊇ 1X . A function f : X → X is a single-valued, total relation.
These notions extend naturally to relations R ⊆ X × Y .
6.1.3
Involutions
A fixed-point free involution on a set X is a function f : X → X such
that
f 2 = 1X ,
f ∩ 1X = ∅.
Thus for such a function f (x) = y ⇔ x = f (y) and f (x) 6= x. We write
Inv(X) for the set of fixed-point free involutions on a set X. Note that
Inv(X) is not closed under function composition; nor does it contain
the identity function. We must look elsewhere for suitable notions of
composition and identity.
An involution is equivalently described as a parition of X into 2element subsets:
[
X=
E,
where E = {{x, y} | f (x) = y}.
(3)
This defines an undirected graph Gf = (X, E). Clearly Gf is 1-regular
[18]: each vertex has exactly one incident edge. Conversely, every graph
G = (X, E) with this property determines a unique f ∈ Inv(X) with
Gf = G. Note that a finite set can only carry such a structure of its
cardinality is even.
6.2
Formalizing diagrams
From our previous discussion, it is fairly clear how we will proceed to
formalize morphisms n → m in D. Given n, m ∈ N, we define N(n, m) =
[n] ⊕ [m]. We visualize this partial order as
1
2
1′
2′
···
n
··· ···
m′
We use the notation i′ to distinguish the elements of [m] in this disjoint
union from those of [n], which are unprimed. Note that the order on
N(n, m) has an immediate spatial interpretation in the diagrammatic
representation: i < j just in case i lies to the left of j on either the top
or bottom line of dots, corresponding to [n] and [m] respectively.
A diagram connecting up dots pairwise will be formalized as a map
f ∈ Inv(|N(n, m)|). Such a map can be visualized by drawing undirected
arcs between the pairs of nodes i, j such that f (i) = j.
6.2.1
Example
The map f ∈ Inv(|N(4, 2)|) such that
f : 1 ↔ 2′ ,
is depicted thus:
1
3 ↔ 1′
2 ↔ 4,
2
3
4
1′ 2′
Our task is now is characterize those involutions which are planar.
The key idea is that this can be done using just the order relations we
have introduced.
6.3
Characterizing Planarity
A map f ∈ Inv(|N(n, m)|) will be called planar if it satisfies the following two conditions, for all i, j ∈ N(n, m):
(PL1)
i < j < f (i)
=⇒ i < f (j) < f (i)
(PL2) f (i) # i < j # f (j) =⇒
f (i) < f (j).
It is instructive to see which possibilities are excluded by these conditions.
6.3.1
First condition
(PL1) i < j < f (i) =⇒ i < f (j) < f (i)
(PL1) rules out
···
f (j)
i
···
j f (i)
where f (j) # f (i), and also
i
j
f (i) f (j)
where f (i) < f (j).
6.3.2
Second condition
(PL2) f (i) # i < j # f (j) =⇒ f (i) < f (j).
Similarly, (PL2) rules out
···
f (j)
i
···
···
j
f (i)
We write P(n, m) for the set of planar maps in Inv(|N(n, m)|).
PROPOSITION 1.6
1. Every planar diagram satisfies the two conditions.
2. Every involution satisfying the two conditions can be drawn as a
planar diagram.
Rather than proving this directly, it is simpler, and also instructive, to
reduce it to a special case. We consider arrows in D of the special form
I → n. Such arrows consist only of caps. They correspond to points, or
states in the terminology of Section 5.
Since the top row of dots is empty, in this case we have a linear order,
and the premise of condition (PL2) can never arise. Hence planarity
for such arrows is just the simple condition (PL1) — which can be seen
to be equivalent to saying that, if we write a left parenthesis for each
left end of a cap, and a right parenthesis for each right end, we get a
well-formed string of parentheses. Thus
corresponds to
()(()).
(Of course, exactly similar comments apply to arrows of the form n → I,
i.e. costates.) It is also clear5 that Proposition 1.6 holds for such arrows.
Now we recall that quite generally, in any pivotal category we have
the Hom-Tensor adjunction
A ⊗ B∗
⌊f ⌋
- I
≃
←→
A
pf q = (1A∗ ⊗ f ) ◦ ηA : I → A∗ ⊗ B
f
- B
≃
←→
I
⌈f ⌉
- A∗ ⊗ B
xf y = ǫB ◦ (f ⊗ 1B ∗ ) : A ⊗ B ∗ → I.
We call pf q the name of f , and xf y the coname. The inverse to the
map f 7→ pf q is defined by
g : I → A∗ ⊗ B 7→ (ǫA ⊗ 1B ) ◦ (1A ⊗ g) : A → B.
For example, we compute the name of the left wave:
=
Applying the inverse transformation:
=
Note also that the unit is the name of the identity: ηn = p1n q, and
similarly ǫn = x1n y.
Thus we see that diagrammatically, the process of forming the name
of an arrow involves reversing the left-right order of the top row of dots
by rotating them concentrically, and sliding them down to lie parallel
with, and to the left of, the bottom row. In this process, cups are turned
5 With
an implicit appeal to the Jordan Curve Theorem!
into caps, while through lines are stretched out and turned to also form
caps.
This transposition of the top row of dots can be described ordertheoretically, as replacing the partial order [n] ⊕ [m] by the linear order
[n]op [m]. Note that the underlying sets of these two partial orders are
the same: |[n] ⊕ [m]| = |[n]op [m]|. Thus pf q is essentially the same
function as f .
PROPOSITION 1.7
For f ∈ Inv(|N(n, m)|), the following are equivalent:
1. f satisfies (PL1) and (PL2) with respect to [n] ⊕ [m].
2. f satisfies (PL1) with respect to [n]op [m].
Proof
Firstly, assume (2), and suppose f (i) # i < j # f (j). If
i < j in the bottom row, then f (j) < i < j in [n]op [m], so by (PL1),
f (j) < f (i) < j, i.e. f (i) < f (j) in [n] ⊕ [m], as required. Now suppose
i < j in the top row. Then j < i < f (j) in [n]op [m], so by (PL1),
j < f (i) < f (j), and in particular f (i) < f (j).
Now assume (1), and suppose that i < j < f (i) in [n]op [m]. The
interesting case is where i is in the top row and f (i) in the bottom
row. We need to do some case analysis. Suppose firstly that j is in the
top row. If f (j) is in the bottom row, then f (j) # j < i # f (i) in
[n] ⊕ [m], and we can apply (PL2) to conclude that f (j) < f (i), and
hence i < f (j) < f (i) in [n]op [m]. If f (j) is in the top row, we must
have f (j) < i by (PL1) for [n] ⊕ [m], and hence i < f (j) < f (i) in
[n]op [m].
Now suppose that j is in the bottom row. If f (j) is in the bottom
row, we must have f (j) < f (i) by (PL1). If f (j) is in the top row, then
we have f (j) # j < f (i) # i in [n] ⊕ [m], so by (PL2) we have f (j) < i,
and hence i < f (j) < f (i) in [n]op [m].
Since (PL1) characterizes planarity for pf q, it follows that (PL1) and
(PL2) characterize planarity for f .
6.4
The Temperley-Lieb Category
Our aim is now to define a category TL, which will yield the desired
description of the diagrammatic category D. The objects of TL are the
natural numbers. The homset TL(n, m) is defined to be the cartesian
product N × P(n, m). Thus a morphism n → m in TL consists of a pair
(k, f ), where k is a natural number, and f ∈ P(n, m) is a planar map in
Inv(|N(n, m)|). The idea is that k is a counter for the number of loops,
so such an arrow can be written δ k · f in the notation used previously.
It remains to define the composition and identities in this category.
Clearly (even leaving aside the natural number components of morphisms) composition cannot be defined as ordinary function composition. This does not even make sense — the codomain of an involution
f ∈ P(n, m) does not match the domain of an involution g ∈ P(m, p) —
let alone yield a function with the necessary properties to be a morphism
in the category.
6.4.1
Composition: The “Execution Formula”
Consider a map f : [n]+[m] −→ [n]+[m]. Each input lies in either [n]
or [m] (exclusive or), and similarly for the corresponding output. This
leads to a decomposition of f into four disjoint partial maps:
fn,n : [n] −→ [n]
fn,m : [n] −→ [m]
fm,n : [m] −→ [n]
fm,m : [m] −→ [m]
so that f can be recovered as the disjoint union of these four maps. If
f is an involution, then these maps will be partial involutions.
Note that these components have a natural diagrammatic reading:
c
the through
fn,n describes the cups of f , fm,m the caps, and fn,m = fm,n
lines.
Now suppose we have maps f : [n]+[m] → [n]+[m] and g : [m]+[p] →
[m] + [p]. We write the decompositions of f and g as above in matrix
form:




gm,m gm,p
fn,n fn,m


g=
f =
gp,m gp,p
fm,n fm,m
We can view these maps as binary relations on [n] + [m] and [m] + [p]
respectively, and use relational algebra (union R ∪ S, relational composition R; S and reflexive transitive closure R∗ ) to define a new relation
θ on [n] + [p]. If we write


θn,n θn,p

θ=
θp,n θp,p
so that θ is the disjoint union of these four components, then we can
define it component-wise as follows:
θn,n = fn,n ∪ fn,m ; gm,m ; (fm,m ; gm,m )∗ ; fm,n
θn,p = fn,m ; (gm,m ; fm,m )∗ ; gm,p
θp,n = gp,m ; (fm,m ; gm,m )∗ ; fm,n
θp,p
= gp,p ∪ gp,m ; fm,m ; (gm,m ; fm,m )∗ ; gm,p .
We can give clear intuitive readings for how these formulas express composition of paths in diagrams in terms of relational algebra:
• The component θn,n describes the cups of the diagram resulting
from the composition. These are the union of the cups of f (fn,n ),
together with paths that start from the top row with a through
line of f , given by fn,m , then go through an alternating odd-length
sequence of cups of g (gm,m ) and caps of f (fm,m ), and finally
return to the top row by a through line of f (fm,n ).
···
• Similarly, θp,p describes the caps of the composition.
c
• θn,p = θp,n
describe the through lines. Thus θn,p describes paths
which start with a through line of f from n to m, continue with
an alternating even-length (and possibly empty) sequence of cups
of g and caps of f , and finish with a through line of g from m to
p.
···
All through lines from n to p must have this form.
This formula corresponds to the interpretation of Cut-Elimination in the
Geometry of Interaction interpretation of proofs in Linear Logic (and by
extension in related logics and type theories) [23]. A more abstract and
general perspective on how this construction arises can be given in the
setting of traced monoidal categories [27, 1].
PROPOSITION 1.8
If f and g are planar, so is θ.
We write θ = g ⊙ f ∈ P(n, p)
6.4.2
Cycles
Given f ∈ P(n, m), g ∈ P(m, p), we define χ(f, g) := fm,m ; gm,m .
Note that χ(f, g)c = (gm,m ; fm,m ), and
χ(f, g); χ(f, g)c ⊆ 1[m] ,
χ(f, g)c ; χ(f, g) ⊆ 1[m] .
Thus χ(f, g) is a partial bijection. However, in general it is neither an
involution, nor fixpoint-free. The cyclic elements of χ(f, g) are those
elements of [m] which lie in the intersection
χ(f, g)+ ∩ 1[m] .
···
Thus if i is a cyclic element, there is a least k > 0 such that χ(f, g)k (i) =
i. The corresponding cycle is
{i, χ(f, g)(i), . . . , χ(f, g)k−1 (i)}.
Distinct cycles are disjoint. We write Z(f, g) for the number of distinct
cycles of χ(f, g).
6.4.3
Composition and Identities
Finally, we define the composition of morphisms in TL. Given (s, f ) :
n → m and (t, g) : m → p:
(t, g) ◦ (s, f ) = (s + t + Z(f, g), g ⊙ f ).
The identity morphism idn : n → n is defined to be the pair (0, τn,n ),
where τn,n is the twist map on [n] + [n]; i.e. the involution i ↔ i′ .
Diagrammatically, this is just
1 2
n
···
1′ 2′
···
n′
Note that this is not the identity map on [n] + [n] — indeed it is (necessarily) fixpoint free!
PROPOSITION 1.9
TL with composition and identities as defined above is a category.
6.4.4
TL as a pivotal category
The monoidal structure of TL is straightforward. If (k, f ) : n → m
and (l, g) : p → q, then (k + l, f + g) : n + p → m + q, where f + g ∈
P(n + p, m + q) is the evident disjoint union of the involutions f and g.
The unit ηn : I → n + n is given by
i ↔ i′
(1 ≤ i ≤ n),
and similarly for the counit. Note that identities, units and counits are
all essentially the same maps, but with distinct types, which partition
their arguments between inputs and outputs differently.
We describe the dual, adjoint and conjugate of an arrow (k, f ) : n →
∼
=
m. Let τn,m : [n] + [m] −→ [m] + [n] be the symmetry isomorphism of
the disjoint union, and
∼
=
ρn : [n] −→ [n] :: i 7→ n − i + 1
be the order-reversal isomorphism. Note that
−1
τn,m
= τm,n ,
ρ−1
n = ρn ,
τn,m ◦ (ρn + ρm ) = (ρm + ρn ) ◦ τn,m .
Then we have (k, f )• = (k, f • ), where:
−1
f † = τn.m ◦ f ◦ τn,m
f∗ = (ρn + ρm ) ◦ f ◦ (ρn + ρm )−1
f ∗ = (f † )∗ .
6.4.5
The Main Result
THEOREM 1.1
TL is isomorphic as a strict, pivotal dagger category to D.
As an immediate Corollary of this result and Proposition 1.2, we have:
THEOREM 1.2
TL is the free strict, pivotal dagger category on one self-dual generator.
This is in the same spirit as the characterizations of free compact and
dagger compact categories in [34, 3].
These results can easily be extended to descriptions of the free pivotal
dagger category over an arbitrary generating category, leading to oriented Temperley-Lieb algebras with primitive (physical) operations. We
refer to [3] for a more detailed presentation (in the symmetric case).
7
Planar λ-Calculus
Our aim in this section is to show how a restricted form of λ-calculus
can be interpreted in the Temperley-Lieb category, and how β-reduction
of λ-terms, which is an important foundational paradigm for computation, is then reflected diagrammatically as geometric simplification,
i.e. “yanking lines straight”. We can give only a brief indication of what
is in fact a rich topic in its own right. See [6, 7, 31] for discussions of
related matters.
7.1
The λ-Calculus
We begin with a (very) brief review of the λ-calculus [14, 9], which
is an important foundational paradigm in Logic and Computation, and
in particular forms the basis for all modern functional programming
languages.
The syntax of the λ-calculus is beguilingly simple. Given a set of
variables x, y, z, . . . we define the set of terms as follows:
t ::= x |
tu
|
λx.
|{z}
|{z}t
application abstraction
Notational Convention: We write
t1 t2 · · · tk ≡ (· · · (t1 t2 ) · · ·)tk .
Some examples of terms:
λx. x
λf. λx. f x
λf. λx. f (f x)
λf. λg. λx. g(f (x))
identity function
application
double application
composition g ◦ f
The basic equation governing this calculus is β-conversion:
(λx. t)u = t[u/x]
E.g. (assuming some arithmetic operations are given)
(λf. λx. f (f x))(λx. x + 1)0 = (0 + 1) + 1 = 2.
By orienting this equation, we get a ‘dynamics’ — β-reduction
(λx. t)u → t[u/x]
Despite its sparse syntax, λ-calculus is very expressive—it is in fact a
universal model of computation, equivalent to Turing machines.
7.2
Types
One important way of constraining the λ-calculus is to introduce
Types.
Types are there to stop you doing (bad) things
Types are in fact one of the most fruitful positive ideas in Computer
Science!
We shall introduce a (highly restrictive) type system, such that the
typable terms can be interpreted in the Temperley-Lieb category (in
fact, in any pivotal category).
Firstly, assuming some set of basic types B, we define a syntax of
general types:
T ::= B | T → T.
Intuitively, T → U represents the type of functions which take inputs of
type T to outputs of type U .
Notational Convention: We write
T1 → T2 → · · · Tk → Tk+1
≡
T1 → (T2 → · · · (Tk → Tk+1 ) · · ·).
Examples:
A→A→A
(A → A) → A
first-order function type
second-order function type
We now introduce a formal system for deriving typing judgements, of
the form:
x1 : T1 , . . . xk : Tk ⊢ t : T.
Such a judgement asserts that the term t has type T under the assumption (or: in the context ) that the variable x1 has type T1 , . . . , xk has
type Tk . All the variables xi appearing in the context must be distinct
— and in our setting, the order in which the variables appear in the list
is significant.
There is one basic form of axiom, for typing variables:
Variable
x:T ⊢x:T
and two inference rules, for typing abstractions and applications respectively:
Function
Γ, x : U ⊢ t : T
Γ ⊢ λx. t : U → T
Γ⊢t:U →T
∆⊢u:U
Γ, ∆ ⊢ tu : T
Note that Γ, ∆ represents the concatenation of the lists Γ, ∆. This implies that the variables appearing in Γ and ∆ are distinct—an important
linearity constraint in the sense of Linear Logic [22].
7.3
Interpretation in Pivotal Categories
We now show how terms typable in our system can be interpreted in a
pivotal category C. We assume firstly that the basic types B have been
interpreted as objects JBK of C. We then extend this to general types
by:
JT → U K = JU K ⊗ JT K∗ .
Now we show how, for each typing judgement Γ ⊢ t : T , to assign an
arrow
JΓK −→ JT K,
where if Γ = x1 : T1 , . . . xk : Tk ,
JΓK = JT1 K ⊗ · · · ⊗ JTk K.
This assignment is defined by induction on the derivation of the typing
judgement in the formal system.
Variable
x:T ⊢x:T
1JT K : JT K −→ JT K
Abstraction
To interpret λ-abstraction, we use the adjunction
Λr : C(A ⊗ B, C) ≃ C(A, C ⊗ B ∗ )
Λr (f ) = A
1A ⊗ ηB∗
-
A ⊗ B ⊗ B∗
∗
f ⊗ 1BC ⊗ B∗
We can then define:
Γ, x : U ⊢ t : T
Γ ⊢ λx : U. t : U → T
JtK : JΓK ⊗ JU K −→ JT K
Λr (JtK) : JΓK −→ JT K ⊗ JU K∗
Application
We use the following operation of right application:
RApp : C(C, B ⊗ A∗ ) × C(D, A) −→ C(C ⊗ D, B)
RApp(f, g) = C ⊗ D
∗
f ⊗g
1B ⊗ ǫBB.
B ⊗ A∗ ⊗ A
We can then define:
Γ⊢t:U →T
∆⊢u:U
Γ, ∆ ⊢ tu : T
JtK : JΓK −→ JT K ⊗ JU K∗
JuK : J∆K −→ JU K
RApp(JtK, JuK) : JΓK ⊗ J∆K −→ JT K
It can be proved that this interpretation is sound for β-conversion, i.e.
J(λx. t)uK = Jt[u/x]K
in any pivotal category.
7.4
An Example
We now discuss an example to show how all this works diagrammatically in TL. We shall consider the bracketing combinator
B ≡ λx.λy.λz. x(yz).
This is characterized by the equation
Babc = a(bc).
Firstly, we derive a typing judgement for this term:
y:A→B⊢y:A→B
z:A⊢z:A
x:B→C⊢x:B→C
y : A → B, z : A ⊢ yz : B
x : B → C, y : A → B, z : A ⊢ x(yz) : C
x : B → C, y : A → B ⊢ λz. x(yz) : A → C
x : B → C ⊢ λy.λz. x(yz) : (A → B) → (A → C)
⊢ λx.λy.λz. x(yz) : (B → C) → (A → B) → (A → C)
Now we take A = B = C = 1 in TL. The interpretation of the open
term
x : B → C, y : A → B, z : A ⊢ x(yz) : C
is as follows:
+
−
x+ x− y y z +
o
Here x is the output of x, and x− the input, and similarly for y. The
output of the whole expression is o. When we abstract the variables, we
obtain the following caps-only diagram:
+
o z + y − y + x− x+
Now we consider an application Babc:
a
+ − + − +
oz y y x x
7.5
b
c
=
a
b
c
o
Discussion
The typed λ-calculus we have used here is in fact a fragment of the
Lambek calculus [36], a basic non-commutative logic and λ-calculus,
which has found extensive applications in computational linguistics [13,
38]. The Lambek calculus can be interpreted in any monoidal biclosed
category, and has notions of left abstraction and application, as well as
the right-handed versions we have described here. Pivotal categories
have stronger properties than monoidal biclosure; for example, duality
and adjoints allow the left- and right-handed versions of abstraction and
application to be defined in terms of each other. Moreover, the duality
means that the corresponding logic has a classical format, with an involutive negation. Thus there is much more to this topic than we have had
the time to discuss here. We merely hope to have given an impression of
how the geometric ideas expressed in the Temperley-Lieb category have
natural connections to a central topic in Logic and Computation.
8
Further Directions
We hope to have given an indication of the rich and suggestive connections which exist between ideas stemming from knot theory, topology
and mathematical physics, on the one hand, and logic and computation
on the other, with the Temperley-Lieb category serving as an intuitive
and compelling meeting point. We hope that further investigation will
uncover deeper links and interplays, leading to new insights in both
directions.
We conclude with a few specific directions for future work:
• The symmetric case, where we drop the planarity constraint, is also
interesting. The algebraic object corresponding to the TemperleyLieb algebra in this case is the Brauer algebra [12], important
in the representation theory of the Orthogonal group (Schur-Weyl
duality). Indeed, there are now a family of various kinds of diagram
algebras: partition algebras, rook algebras etc., arising in quantum
statistical mechanics, and studied in Representation Theory [25].
• The categorical perspective suggests oriented versions of the TemperleyLieb algebra and related structures, where we no longer have A =
A∗ . This is also natural from the point of view of Quantum
Mechanics, where this non-trivial duality on objects distinguishes
complex from real Hilbert spaces.
• We can ask how expressive planar Geometry of Interaction is; and
what rôle may be played by braiding or other geometric information.
• Again, it would be interesting to understand the scope and limits of
planar Quantum Mechanics and Quantum information processing.
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